Analytical and dynamical insights into the (2 + 1) dimensional fractional mKdV Zakharov Kuznetsov framework

Fractional nonlinear evolution equations serve as a fundamental model to capture complex phenomena with memory effects, anomalous dispersion and long-range interaction. In this study, the (2 + 1)-dimensional fractional modified Korteweg–de Vries–Zakharov–Kuznetsov (mKdV–ZK) equation is studied, and its application in the context of nonlinear wave propagation in plasma physics and other nonlinear media is discussed. A wide range of exact analytical solutions is successfully generated by using the improved modified extended tanh-expansion method, which involves bright solitons, dark solitons, kink waves, and periodic wave profiles. The acquired solutions show rich nonlinear behaviors originating from the interaction between the fractional-order characteristics and the transverse dispersion parameters. By using an appropriate fractional traveling wave transformation, the governing model is reduced to a nonlinear dynamical system, which allows a comprehensive qualitative analysis of wave dynamics. This is done through phase portraits, bifurcation structure analysis, Jacobian eigenvalue analysis, and sensitivity investigations of the dynamical properties of the system. The results show that fractional parameters, even when changed slightly, may change stability characteristics and cause transitions between different types of periodic, quasi-periodic and chaotic wave motions. The study relates in particular the exact wave solutions to the global dynamics underlying this model, a connection that has not been investigated well enough in the previous studies for the fractional mKdV–ZK model. A unified analytical and dynamical framework is introduced in this work, which gives new insights into the influence of space–time fractional effects on the evolution of nonlinear waves and offers a general framework for the understanding of more complex wave phenomena in higher-dimensional fractional systems of nonlinear waves.